Conjecture on the complexity of products of powers of 2 and 3

Let $\|n\|$ denote the smallest number of ones needed to write the positive integer $n$ using addition and multiplication. For integers $k,\ell\geq 0$, with $k$ and $\ell$ not both equal to $0$, the integer under consideration is $2^k3^\ell$. Integer-complexity conjecture.

$$\|2^k3^\ell\|=2k+3\ell.$$

This combines the known equality $\|3^\ell\|=3\ell$ for $\ell\geq 1$ with the conjectured equality $\|2^k\|=2k$ for $k\geq 1$. The claim was verified in the paper for $k\leq 48$ and arbitrary $\ell$, excluding $k=\ell=0$, but remains open in general.